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A357465
Decimal expansion of the real root of 3*x^3 - x^2 - 1.
2
8, 2, 4, 1, 2, 2, 6, 2, 1, 1, 0, 9, 1, 3, 2, 9, 6, 6, 3, 1, 2, 2, 7, 8, 9, 7, 9, 8, 7, 0, 2, 8, 2, 5, 6, 2, 6, 4, 3, 3, 2, 6, 4, 1, 4, 3, 7, 0, 6, 3, 8, 7, 2, 8, 9, 1, 6, 0, 4, 3, 7, 6, 5, 4, 2, 0, 9, 7, 8, 0, 9, 8, 6, 8, 1, 2
OFFSET
0,1
COMMENTS
This equals r0 + 1/9 where r0 is the real root of y^3 - (1/27)*y - 245/729.
The other roots of 3*x^3 - x^2 - 1 are (w1*(4*(245 + 9*sqrt(741)))^(1/3) + w2*(4*(245 - 9*sqrt(741)))^(1/3) + 2)/18 = -0.2453946438... + 0.5867299404...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (1 - cosh((1/3)*arccosh(245/2)) + sqrt(3)*sinh((1/3)*arccosh(245/2))*i)/9 and its complex conjugate.
FORMULA
r = ((4*(245 + 9*sqrt(741)))^(1/3) + 4*(4*(245 + 9*sqrt(741)))^(-1/3) + 2)/18.
r = ((4*(245 + 9*sqrt(741)))^(1/3) + (4*(245 - 9*sqrt(741)))^(1/3) + 2)/18.
r = (1 + 2*cosh((1/3)*arccosh(245/2)))/9.
EXAMPLE
0.824122621109132966312278979870282562643326414370638728916043765420978098...
MATHEMATICA
RealDigits[x /. FindRoot[3*x^3 - x^2 - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 07 2022 *)
CROSSREFS
Cf. A357464.
Sequence in context: A298571 A096257 A333206 * A125578 A083729 A019775
KEYWORD
nonn,cons,easy
AUTHOR
Wolfdieter Lang, Sep 30 2022
STATUS
approved